It depends where in the UK they are and if they are connected to the church or are one of the town/city run ( or privately owned) cemeteries.
If the large municiple 'generally' they are run by the local council and many have search facilities on their council website, some free, some you have to pay to get details of who is buried where. Same with private, contact them via their website for details, free/fee will depend on each. Many have little to no space now for burials so only have cremations, so there is no head stone, some has a book of rememberance which is often under glass with the page opened for that date of details of who was cremated...so go on a different date and you will not be able to see and will have to pay for details from the council offices.
I used to live near to a large cemetery ( and used it as a short cut to school) My grandparents are buried there, she in the 1940s ( when my dad was 4yrs old) and my granddad in the 1960s. I could never find them until I contacted the council and got a grave number /plan even then it took a long time and all that is there is a cracked flat stone with a number on it. There is ground staff there cutting grass and lifting 'momentos' left on graves....and there is certainly a different 'feeling' between new/old now grave stones have to be agreed, size, colour, style and are very standard in straight tight rows, no grave boundary kerb and if you turn and look at the old part you see complete contrast with large, ornate stone wear.
From the councils website:
"You can buy a grave plot in a Leicester cemetery for a period of 50 years, renewable at the end of the lease." So realistically most are no renewed after 50yrs so up to the council what they do as it reverts back to them.
My cousin lives next door to the village church as far back as I can remember granddad, then uncle, now cousin cut the grass in the church yrd ( which is full) Last time I visited the church yrd was almost empty of stones and many lined the boundary wall.... I was told any damaged/unsafe headstones were removed as the church is liable if they fell and hurt someone, so H&S (although it makes for easy grass cutting)
My grandfather gave the church an acre of field just outside the village to use and that is the second church yrd and certainly for the last 30 yrs only cremations no burials although each has a small inscribed plinth.
There are many cemeteries who have 'friends of xxx cemetery' who photograph gravestones and several have websites eg
http://www.fowrcl.org.uk/ Findagrave used to be mainly US graves, it seems now lots of grave websites often run by individuals or 'friends of' have increased their site by adding theirs, there are several fee ones now in the UK too.
The difference between US and UK is that in the UK burials are not connected to death certs so for FH that can make it harder in finding where people are.
So like my dad who died in Derby hospital is registered in Derby as died, yet buried ( well cremated) in Leicester and there is no connecting records that show that, so someone in the future looking for my dad will find it hard to follow although there is the freebmd index to help